It has been established that the programmed storage and dispensing of medicines or drugs through the use of a small portable dispenser such as disclosed in my U.S. patents identified above, as well as my copending U.S. patent application identified above, can significantly increase the chances that the medicines will be effective for the intended purpose. Since such dispensers are intended for home or personal use, it has now become desirable to safeguard them against access to the medicine compartments by infants or children. Indeed, it is understood that in some countries laws or regulations have been promulgated requiring medicine dispensers to contain some sort of mechanism for either preventing unauthorized access to the medicines by children or for at least providing an impediment to the easy, simple opening of the medicine dispenser by children. Therefore, the present invention in one aspect is addressed to the aforementioned problem in the context of medicine dispensers of the type disclosed in my aforementioned patents and pending patent application but it will be apparent as the description proceeds that the invention will have applicability to other forms of medicine dispensers of the same general character.